In the last decade, a number of public key cryptosystems based on com- binatorial group theoretic problems in braid groups have been proposed. We survey these and some known attacks them. This includes: Basic facts Garside normal form its elements, algorithms for solving word problem group, major public-key cryptosystems. conclude with discussion future directions (which includes also description which are other non-commutative groups).

Using Reiner's definition of Stirling numbers the second kind in types $B$ and $D$, we generalize two well-known identities concerning classical kind. The first identity relates them with Eulerian interprets as entries a transition matrix between elements standard bases polynomial ring $\mathbb{R}[x]$. Finally, these to group colored permutations $G_{m,n}$.

Abstract The Anshel–Anshel–Goldfeld (AAG) key-exchange protocol was implemented and studied with the braid groups as its underlying platform. length-based attack, introduced by Hughes Tannenbaum, has been used to cryptanalyze AAG in this setting. Eick Kahrobaei suggest use polycyclic a possible platform for protocol. In paper, we apply several known variants of attack against group experimental results show that, these groups, are unsuccessful case having high Hirsch length. This suggests...

We generalize the results of Ksavrelof and Zeng about multidistribution excedance number $S_n$ with some natural parameters to colored permutation group Coxeter type $D$. define two different orders on these groups which induce numbers. Surprisingly, in case group, we get same generalized formulas for both orders.

A conjugation-free geometric presentation of a fundamental group is with the natural topological generators x 1 ,…,x n and cyclic relations: [Formula: see text] no conjugations on generators. We have already proved in [13] that if graph arrangement disjoint union cycles, then its has presentation. In this paper, we extend property to arrangements whose graphs are cycle-tree graphs. Moreover, study some properties type presentations for line arrangement's complement. show these satisfy...

Wiring diagrams usually serve as a tool in the study of arrangements lines and pseudolines. Here we go opposite direction, using known properties line to motivate certain equivalence relations actions on sets wiring diagrams, which preserve incidence lattice fundamental groups affine projective complements diagrams. These are used [GTV] classify real up 8 show that this case, determines both groups.

The braid group Bn, endowed with Artin's presentation, admits an antiautomorphism Bn → such that v is defined by reading braids in reverse order (from right to left instead of right).We prove the map vv injective.We also give some consequences arising due this injectivity.

Using Reiner's definition of Stirling numbers type B the second kind, we provide a 'balls into urns' approach for proving generalization well-known identity concerning classical kind: $x^n=\sum\limits_{k=0}^n{S(n,k)[x]_k}.$

The group of alternating colored permutations is the natural analogue classical group, inside wreath product $\mathbb{Z}_r \wr S_n$. We present a 'Coxeter-like' presentation for this and compute length function with respect to that presentation. Then, we as covering $\mathbb{Z}_{\frac{r}{2}} S_n$ use point view give another expression function. also lift several known parameters permutations.

The excedance number for S_n is known to have an Eulerian distribution. Nevertheless, the classical proof uses descents rather than excedances. We present a direct recursive which seems be folklore and extend it colored permutation groups G_r,n. generalized recursion yields some interesting connection Stirling numbers of second kind. also show logconcavity result concerning variant number. Finally, we that generating function defined on G_r,n symmetric.

We define an excedance number for the multi-colored permutation group i.e. wreath product $({\Bbb Z}_{r_1} \times \cdots {\Bbb Z}_{r_k}) \wr S_n$ and calculate its multi-distribution with some natural parameters. also compute multi–distribution of parameters exc$(\pi)$ fix$(\pi)$ over sets involutions in group. Using this, we count this having a fixed excedances absolute points.

We define new statistics, (c, d)-descents, on the colored permutation groups Z_r \wr S_n and compute distribution of these statistics elements in groups. use some combinatorial approaches, recurrences, generating functions manipulations to obtain our results.